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ZF and its interpretations Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-03-04 S. Jockwich Martinez, S. Tarafder, G. Venturi
In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for = and ∈. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of to new algebra-valued models. This paper presents, for the first time, non-trivial
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A good lightface [formula omitted] well-ordering of the reals does not imply the existence of boldface [formula omitted] well-orderings Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-02-28 Vladimir Kanovei, Vassily Lyubetsky
We make use of a finite support product of the Jensen-type forcing notions to define a model of the set theory in which, for a given , there exists a good lightface well-ordering of the reals but there are no any (not necessarily good) well-orderings in the boldface class .
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Nonstandard proof methods in toposes Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-02-22 José Siqueira
We determine sufficient structure for an elementary topos to emulate Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and maps. This development allows one to employ the proof methods of nonstandard analysis (transfer, standardisation, and idealisation) in new environments such as toposes
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A New Model Construction by Making a Detour via Intuitionistic Theories IV: A Closer Connection between KPω and BI Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-02-15 Kentaro Sato
By combining tree representation of sets with the method introduced in the previous papers in the series, we give a new interpretation of (Kripke–Platek set theory with the foundation schema restricted to augmented by ) in for any sentence , in such a way that sentences are preserved, where the language of second order arithmetic is considered as a sublanguage of that of set theory via the standard
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Arboreal categories and equi-resource homomorphism preservation theorems Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-02-15 Samson Abramsky, Luca Reggio
The classical homomorphism preservation theorem, due to Łoś, Lyndon and Tarski, states that a first-order sentence is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence . Given a notion of (syntactic) complexity of sentences, an “equi-resource” homomorphism preservation theorem improves on the classical result by ensuring that can
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Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-02-01 Aristotelis Panagiotopoulos, Assaf Shani
The algebraic dimension of a Polish permutation group is the size of the largest with the property that the orbit of every under the pointwise stabilizer of is infinite. We study the Bernoulli shift for various Polish permutation groups and we provide criteria under which the -shift is generically ergodic relative to the injective part of the -shift, when has algebraic dimension ≤. We use this to show
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The formal verification of the ctm approach to forcing Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-01-30 Emmanuel Gunther, Miguel Pagano, Pedro Sánchez Terraf, Matías Steinberg
We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model of , of generic extensions satisfying and . Moreover, let be the set of instances of the Axiom of Replacement. We isolated a 21-element subset and defined such that for every and -generic , implies , where is Zermelo set theory with Choice.
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Sharp Vaught's conjecture for some classes of partial orders Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-01-12 Miloš S. Kurilić
Matatyahu Rubin has shown that a sharp version of Vaught's conjecture, I(T,ω)∈{0,1,c}, holds for each complete theory of linear order T. We show that the same is true for each complete theory of partial order having a model in the minimal class of partial orders containing the class of linear orders and which is closed under finite products and finite disjoint unions. The same holds for the extension
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Towards characterizing the >ω2-fickle recursively enumerable Turing degrees Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2024-01-05 Liling Ko
Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees 〈RT,≤T〉, it is not known how one can characterize the degrees d∈RT below which L can be embedded. Two important characterizations are of the L7 and M3 lattices, where the lattices are embedded below d if and only if d contains sets of “fickleness” >ω and ≥ωω respectively. We work towards finding a lattice
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Structural and universal completeness in algebra and logic Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-12-16 Paolo Aglianò, Sara Ugolini
In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural
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SOP1, SOP2, and antichain tree property Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-12-11 JinHoo Ahn, Joonhee Kim
In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP2 can be witnessed by a formula with a tree of tuples holding ‘arbitrary homogeneous inconsistency’ (e.g., weak k-TP1 conditions or other possible inconsistency configurations). And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP1, and by using this, we investigate
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Constructing the constructible universe constructively Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-12-07 Richard Matthews, Michael Rathjen
We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic Kripke-Platek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive Zermelo-Fraenkel (even
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On categorical structures arising from implicative algebras: From topology to assemblies Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-11-20 Samuele Maschio, Davide Trotta
Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras
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Probabilistic temporal logic with countably additive semantics Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-11-14 Dragan Doder, Zoran Ognjanović
This work presents a proof-theoretical and model-theoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A
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On duality and model theory for polyadic spaces Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-11-07 Sam van Gool, Jérémie Marquès
This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness
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Pathologies in satisfaction classes Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-10-20 Athar Abdul-Quader, Mateusz Łełyk
We study subsets of countable recursively saturated models of PA which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets X such that there is a satisfaction class S where S behaves correctly on an idempotent disjunction of length c if and only if c∈X. We generalize this result to characterize several types of pathologies including double negations
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On the geometric equivalence of algebras Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-10-06 M. Shahryari
It is known that an algebra is geometrically equivalent to any of its filterpowers if it is qω-compact. We present an explicit description for the radicals of systems of equation over an algebra A and then we prove the above assertion by an elementary new argument. Then we define qκ-compact algebras and κ-filterpowers for any infinite cardinal κ. We show that any qκ-compact algebra is geometric equivalent
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Towards a finer classification of strongly minimal sets Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-09-28 John T. Baldwin, Viktor V. Verbovskiy
Let M be strongly minimal and constructed by a ‘Hrushovski construction’ with a single ternary relation. If the Hrushovski algebraization function μ is in a certain class T (μ triples) we show that for independent I with |I|>1, dcl⁎(I)=∅ (* means not in dcl of a proper subset). This implies the only definable truly n-ary functions f (f ‘depends’ on each argument), occur when n=1. We prove for Hrushovski's
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Classification of ℵ0-categorical C-minimal pure C-sets Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-09-27 Françoise Delon, Marie-Hélène Mourgues
We classify all ℵ0-categorical and C-minimal C-sets up to elementary equivalence. As usual the Ryll-Nardzewski Theorem makes the classification of indiscernible ℵ0-categorical C-minimal sets as a first step. We first define solvable good trees, via a finite induction. The trees involved in initial and induction steps have a set of nodes, either consisting of a singleton, or having dense branches without
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Positive modal logic beyond distributivity Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-09-26 Nick Bezhanishvili, Anna Dmitrieva, Jim de Groot, Tommaso Moraschini
We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of Π1-persistence and show that every weak positive modal logic is Π1-persistent. This approach leads to a new
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Weak saturation properties and side conditions Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-09-06 Monroe Eskew
Towards combining “compactness” and “hugeness” properties at ω2, we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at ω2 using Neeman's forcing. On the other hand, we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating
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Generalized fusible numbers and their ordinals Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-09-01 Alexander I. Bufetov, Gabriel Nivasch, Fedor Pakhomov
Erickson defined the fusible numbers as a set F of reals generated by repeated application of the function x+y+12. Erickson, Nivasch, and Xu showed that F is well ordered, with order type ε0. They also investigated a recursively defined function M:R→R. They showed that the set of points of discontinuity of M is a subset of F of order type ε0. They also showed that, although M is a total function on
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Extensions of Solovay's system S without independent sets of axioms Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-09-01 Igor Gorbunov, Dmitry Shkatov
Chagrov and Zakharyaschev posed the problem of existence of extensions of Solovay's system S, which is a non-normalizable quasi-normal modal logic, that do not admit deductively independent sets of axioms. This paper gives a solution by exhibiting countably many extensions of S without deductively independent sets of axioms.
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Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-09-01 Ur Ya'ar
We prove that the theory of the models constructible using finitely many cofinality quantifiers – Cλ1,…,λn⁎ and C<λ1,…,<λn⁎ for λ1,…,λn regular cardinals – is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to <μ-closed
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Indestructibility of some compactness principles over models of PFA Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-08-30 Radek Honzik, Chris Lambie-Hanson, Šárka Stejskalová
We show that PFA (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ω2-Aronszajn tree or a weak ω1-Kurepa tree, and moreover no σ-centered forcing can add a weak ω1-Kurepa tree (a tree of height and size ω1 with at least ω2 cofinal branches). This partially answers an open problem whether ccc forcings can add ω2-Aronszajn trees or ω1-Kurepa trees (with ¬□ω1
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On countably perfectly meager and countably perfectly null sets Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-08-29 Tomasz Weiss, Piotr Zakrzewski
We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set. We say that a subset A of a perfect Polish space X is countably perfectly meager (respectively, countably perfectly null) in X, if for every perfect Polish topology τ on X, giving the original Borel structure of X, A is covered by an Fσ-set F in X with the
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Primitive recursive reverse mathematics Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-08-25 Nikolay Bazhenov, Marta Fiori-Carones, Lu Liu, Alexander Melnikov
We use a second-order analogy PRA2 of PRA to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the fast-developing field of primitive recursive (‘punctual’) algebra and analysis, and with results from ‘online’ combinatorics. We argue that PRA2 is sufficiently robust to serve as an alternative
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Large cardinals at the brink Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-08-10 W. Hugh Woodin
Kunen's theorem that assuming the Axiom of Choice there are no Reinhardt cardinals is a key milestone in the program to understand large cardinal axioms. But this theorem is not the end of a story, rather it is the beginning.
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Counterfactual and seeing-to-it responsibilities in strategic games Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-08-05 Pavel Naumov, Jia Tao
The article studies two forms of responsibility in the setting of strategic games with imperfect information. They are referred to as seeing-to-it responsibility and counterfactual responsibility. It shows that counterfactual responsibility is definable through seeing-to-it, but not the other way around. The article also proposes a sound and complete bimodal logical system that describes the interplay
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The comparison lemma Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-08-05 John R. Steel
The standard comparison lemma of inner model theory is deficient, in that it does not in general produce a comparison of all the relevant inputs. How two mice compare can depend upon which iteration strategies are used to compare them. We shall outline here a method for comparing iteration strategies that removes this defect.
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Towards Logical Foundations for Probabilistic Computation Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-27 Melissa Antonelli, Ugo Dal Lago, Paolo Pistone
The overall purpose of the present work is to lay the foundations for a new approach to bridge logic and probabilistic computation. To this aim we introduce extensions of classical and intuitionistic propositional logic with counting quantifiers, that is, quantifiers that measure to which extent a formula is true. The resulting systems, called cCPL and iCPL, respectively, admit a natural semantics
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A Lindström theorem for intuitionistic first-order logic Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-26 Grigory Olkhovikov, Guillermo Badia, Reihane Zoghifard
We extend the main result of [1] to the first-order intuitionistic logic (with and without equality), showing that it is a maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under asimulations. A similar result is also shown for the intuitionistic logic of constant domains.
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Measurable cardinals and choiceless axioms Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-25 Gabriel Goldberg
Kunen refuted the existence of an elementary embedding from the universe of sets to itself assuming the Axiom of Choice. This paper concerns the ramifications of this hypothesis when the Axiom of Choice is not assumed. For example, the existence of such an embedding implies that there is a proper class of cardinals λ such that λ+ is measurable.
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Locally compact, ω1-compact spaces Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-25 Peter Nyikos, Lyubomyr Zdomskyy
An ω1-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, ω1-compact space is σ-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency
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Some variations on the splitting number Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-24 Saharon Shelah, Juris Steprāns
Variations on the splitting number s are examined by localizing the splitting property to finite sets. To be more precise, rather than considering families of subsets of the integers that have the property that every infinite set is split into two infinite sets by some member of the family a stronger property is considered: Whenever an subset of the integers is represented as the disjoint union of
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Some simple theories from a Boolean algebra point of view Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-24 M. Malliaris, S. Shelah
We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories Tm reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the theories Tn,k, which are the higher-order analogues of the triangle-free random graph. The proof involves building Boolean algebras and ultrafilters “by hand” to satisfy certain
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Alternating (In)Dependence-Friendly Logic Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-22 Dylan Bellier, Massimo Benerecetti, Dario Della Monica, Fabio Mogavero
Hintikka and Sandu originally proposed Independence Friendly Logic ( ) as a first-order logic of imperfect information to describe game-theoretic phenomena underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice game-theoretic semantics in terms of imperfect information
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Causal Modeling Semantics for Counterfactuals with Disjunctive Antecedents Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-20
Causal Modeling Semantics (CMS, e.g., Galles and Pearl, 1998; Pearl, 2000; Halpern, 2000) is a powerful framework for evaluating counterfactuals whose antecedent is a conjunction of atomic formulas. We extend CMS to an evaluation of the probability of counterfactuals with disjunctive antecedents, and more generally, to counterfactuals whose antecedent is an arbitrary Boolean combination of atomic formulas
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Probing the quantitative–qualitative divide in probabilistic reasoning Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-20
This paper explores the space of (propositional) probabilistic logical languages, ranging from a purely ‘qualitative’ comparative language to a highly ‘quantitative’ language involving arbitrary polynomials over probability terms. While talk of qualitative vs. quantitative may be suggestive, we identify a robust and meaningful boundary in the space by distinguishing systems that encode (at most) additive
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Editorial Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-20 Dilip Raghavan
Abstract not available
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Subadditive families of hypergraphs Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-20 Jindřich Zapletal
I analyze a natural class of proper forcings associated with actions of countable groups on Polish spaces, providing a practical and informative characterization as to when these forcings add no independent reals.
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Kunen the expositor Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-18 Akihiro Kanamori
Kunen's expository work is described, bringing out both his way of assimilating and thinking about set theory and how it had a meaningful hand in its promulgation into the next generations.
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Mutually embeddable models of ZFC Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-18 Monroe Eskew, Sy-David Friedman, Yair Hayut, Farmer Schlutzenberg
We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.
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Boolean valued semantics for infinitary logics Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-18 Juan M. Santiago Suárez, Matteo Viale
It is well known that the completeness theorem for Lω1ω fails with respect to Tarski semantics. Mansfield showed that it holds for L∞∞ if one replaces Tarski semantics with Boolean valued semantics. We use forcing to improve his result in order to obtain a stronger form of Boolean completeness (but only for L∞ω). Leveraging on our completeness result, we establish the Craig interpolation property and
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A logico-geometric comparison of coherence for non-additive uncertainty measures Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-17
We investigate the notion of coherence for (non-)additive uncertainty measures from a logico-geometric point of view. Our main result is to the effect that distinct criteria for coherence are not always matched by axiomatically distinct measures of uncertainty. In addition we introduce a metalogic within which this kind of result can be captured formally.
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Asymptotic conditional probabilities for binary probability functions Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-17
The paper investigates the class of probability functions defined on sentences of a predicate language containing binary and possibly also unary predicates which satisfy the Principle of Binary Exchangeability (a strengthening of the Principle of Exchangeability). Following a survey of relevant properties of such functions we prove the main theorems of this paper showing that under some mild conditions
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Probability propagation rules for Aristotelian syllogisms Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-17
We present a coherence-based probability semantics and probability propagation rules for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions
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Reasoning with belief functions over Belnap–Dunn logic Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-17
We design an expansion of Belnap–Dunn logic with belief and plausibility functions that allows non-trivial reasoning with contradictory and incomplete probabilistic information. We also formalise reasoning with non-standard probabilities and belief functions in two ways. First, using a calculus of linear inequalities, akin to the one presented in [23]. Second, as a two-layered modal logic wherein reasoning
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Encoding de Finetti's coherence within Łukasiewicz logic and MV-algebras Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-16
The present paper investigates proof-theoretical and algebraic properties for the probability logic FP(Ł,Ł), meant for reasoning on the uncertainty of Łukasiewicz events. Methodologically speaking, we will consider a translation function between formulas of FP(Ł,Ł) to the propositional language of Łukasiewicz logic that allows us to apply the latter and the well-developed theory of MV-algebras directly
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HL ideals and Sacks indestructible ultrafilters Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-17 David Chodounský, Osvaldo Guzmán, Michael Hrušák
We study ultrafilters on countable sets and reaping families which are indestructible by Sacks forcing. We deal with the combinatorial characterization of such families and we prove that every reaping family of size smaller than the continuum is Sacks indestructible. We prove that complements of many definable ideals are Sacks reaping indestructible, with one notable exception, the complement of the
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On middle box products and paracompact cardinals Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-17 David Buhagiar, Mirna Džamonja
The paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of particular interest are products of the type □<κ2λ, where we prove that for a regular uncountable cardinal κ, if □<κ2λ is paracompact for every λ≥κ, then κ is at least inaccessible
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Zero-dimensional σ-homogeneous spaces Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-16 Andrea Medini, Zoltán Vidnyánszky
All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is σ-homogeneous. Inspired by this theorem, we obtain the following results: • Assuming AD, every zero-dimensional space is σ-homogeneous, • Assuming AC, there exists a zero-dimensional space that is not σ-homogeneous, • Assuming V=L, there exists a coanalytic zero-dimensional space that
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Dense metrizability Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-11 Stevo Todorcevic
If a compact space K has a dense metrizable subspace its regular open algebra forces that the generic ultrafilter is countably generated. We investigate the class of compact spaces K for which the converse of this implication is true and give some applications of this. More precisely, we shall show that this gives us a rather powerful method for proving that a given compact space has a dense metrizable
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Two chain conditions and their Todorčević's fragments of Martin's Axiom Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-10 Teruyuki Yorioka
In this paper, we investigate two chain conditions of forcing notions, called the rectangle refining property and property R1,ℵ1. They are stronger than the countable chain condition. Some of their typical examples are forcing notions about indestructible gaps introduced by Kunen. Both chain conditions are similar and have common examples, however, no distinction between them is known so far. In this
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A Borel maximal eventually different family Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-05 Haim Horowitz, Saharon Shelah
We construct a Borel maximal eventually different family.
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Choice and independence of premise rules in intuitionistic set theory Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-05 Emanuele Frittaion, Takako Nemoto, Michael Rathjen
Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over N. It is also shown that the existence property (or existential definability property) holds for statements of
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The relative strengths of fragments of Martin's axiom Ann. Pure Appl. Logic (IF 0.8) Pub Date : 2023-07-05 Joan Bagaria
We give a survey of results on the relative strengths of different fragments of Martin's Axiom, as well as a list of the main remaining open questions.